There are no linear-phase recursive filters because a recursive filter
cannot generate a symmetric impulse response. However,
it is possible to implement a zero-phase filter offline
using a recursive filter twice. That is, if the entire input signal
is stored in a computer memory or hard disk, for example,
then we can apply a recursive filter both forward and backward in
time. Doing this squares the amplitude response of the filter
and zeros the phase response.

To show this analytically, let
denote the output of the first
filtering operation (which we'll take to be ``forward'' in time in the
normal way), and let
be the impulse response of the recursive
filter. Then we have

where
is the input signal at sample
. For the second pass,
we ``flip''
to obtain
and apply the filter again:

FLIP

The final output is then this result flipped:

FLIP FLIP FLIP FLIP

where the last simplification tells us that flipping the input and
output signals is equivalent to flipping the impulse response
instead. Putting all these operations together, we
have

FLIP FLIPFLIP

By the flip theorem for z transforms, we have that the z transform of
FLIP
is
:

If the filter were complex, then we would need to conjugate its
coefficients when running it backwards.

In summary, we have thus shown that forward-backward
filtering squares the amplitude response and zeros the
phase response.11.3Note also that the phase response is truly zero,
never alternating between zero and
. No matter what nonlinear
phase response
a filter may have, this phase is
completely canceled out by forward and backward filtering. The
amplitude response, on the other hand, is squared. For simple
bandpass filters (including lowpass, highpass, etc.), for which the
desired gain is 1 in the passband and 0 in the stopband, squaring the
amplitude response usually improves the response, because the
``stopband ripple'' (deviation from 0) is squared,
thereby doubling the stopband attenuation in dB. On the other
hand, passband ripple (deviation from 1) is only doubled by the
squaring (because
).

A Matlab example of forward-backward filtering is presented in
§11.6 (in Fig.11.1).